Field theoretical aspects of general relativity - S. SPEZIALE

A concise treatment of some advanced topics in classical gauge theories and GR: use of Hamiltonian formalism and the problem of time in GR, Dirac's algorithm for constrained systems and counting degrees of freedom, applications of Noether's theorem to gauge symmetries and the notion of surface charges. The tetrad formulation of GR and its link to loop quantum gravity and the proposed quantization of space will also be mentioned.
The main goal of the class is to use GR as the main working example to explore the fundamentals of various tools of field theory relevant across disciplines, as well as highlight the own specialties and peculiarites of GR

1. Lagrangian formalism (Maxwell and Einstein-Hilbert Lagrangians; variational principle; role of boundary terms and boundary conditions; first order variational principle; unicity of the Levi-Civita connection)
2. Hamiltonian formalism: electromagnetism (Legendre transform; Gauss law; Dirac's theory of constrained systems; physical phase space and counting of degrees of freedom)
3. Hypersurfaces (Embeddings; Intrinsic and extrinsic curvature; Gauss-Codazzi equations; Gibbons-Hawking boundary term; space-like versus light-like hypersurfaces)
4. Hamiltonian formalism: general relativity (ADM variables; constraint structure and Dirac's hypersurface deformation algebroid; ADM energy)
5. Noether theorem for theories with gauge symmetries (Symmetries and conservation laws; canonical generators; global and local (small and large) gauge symmetries;
surface charges in electromagnetism and gravity)
6. Tetrad variables (Isomorphism between the tangent space and the vectorial bundle; Spin connection; Einstein-Cartan Lagrangian)
7. Hamiltonian formalism: tetrad variables (Constraint structure in tetrad variables; gauge theory phase space; second class constraints; Ashtekar-Barbero-Immirzi canonical transformation)
8. Elements of loop quantum gravity (Wilson loops and spin networks; quanta of space and fuzzy polyhedra)

Basics of classical field theory, electromagnetism and gravity; basics of Hamiltonian mechanics

Maitrise des sujets du cours, pour les étudiants M2 examen oral